Precise asymptotic behavior of strongly decreasing solutions of first-order nonlinear functional differential equations
In this article, we study the asymptotic behavior of strongly decreasing solutions of the first-order nonlinear functional differential equation $$ x'(t)+p(t)| x(g(t))| ^{\alpha -1}x(g(t))=0, $$ where $\alpha $ is a positive constant such that $0<\alpha <1$, p(t) is a positive conti...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2014-10-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2014/206/abstr.html |
| _version_ | 1819070264015585280 |
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| author | George E. Chatzarakis Kusano Takasi Ioannis P. Stavroulakis |
| author_facet | George E. Chatzarakis Kusano Takasi Ioannis P. Stavroulakis |
| author_sort | George E. Chatzarakis |
| collection | DOAJ |
| description | In this article, we study the asymptotic behavior of strongly decreasing
solutions of the first-order nonlinear functional differential equation
$$
x'(t)+p(t)| x(g(t))| ^{\alpha -1}x(g(t))=0,
$$
where $\alpha $ is a positive constant such that $0<\alpha <1$, p(t) is a
positive continuous function on $[a,\infty )$, a>0 and g(t) is a
positive continuous function on $[a,+\infty )$ such that
$\lim_{t\to \infty }g(t)=\infty $. Conditions which guarantee the existence
of strongly decreasing solutions are established, and theorems are stated on the
asymptotic behavior of such solutions, at infinity. The problem it is
studied in the framework of regular variation, assuming that the coefficient
p(t) is a regularly varying function, and focusing on strongly decreasing
solutions that are regularly varying. In addition, g(t) is required to
satisfy the condition
$$
\lim_{t\to \infty }\frac{g(t)}{t}=1.
$$
Examples illustrating the results are also given. |
| first_indexed | 2024-12-21T17:03:10Z |
| format | Article |
| id | doaj.art-93d2881e78a048ef9cbd60fa56dbb400 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-21T17:03:10Z |
| publishDate | 2014-10-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-93d2881e78a048ef9cbd60fa56dbb4002022-12-21T18:56:36ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912014-10-012014206,114Precise asymptotic behavior of strongly decreasing solutions of first-order nonlinear functional differential equationsGeorge E. Chatzarakis0Kusano Takasi1Ioannis P. Stavroulakis2 ASPETE, Athens, Greece Hiroshima Univ., Japan Univ. of Ioannina, Greece In this article, we study the asymptotic behavior of strongly decreasing solutions of the first-order nonlinear functional differential equation $$ x'(t)+p(t)| x(g(t))| ^{\alpha -1}x(g(t))=0, $$ where $\alpha $ is a positive constant such that $0<\alpha <1$, p(t) is a positive continuous function on $[a,\infty )$, a>0 and g(t) is a positive continuous function on $[a,+\infty )$ such that $\lim_{t\to \infty }g(t)=\infty $. Conditions which guarantee the existence of strongly decreasing solutions are established, and theorems are stated on the asymptotic behavior of such solutions, at infinity. The problem it is studied in the framework of regular variation, assuming that the coefficient p(t) is a regularly varying function, and focusing on strongly decreasing solutions that are regularly varying. In addition, g(t) is required to satisfy the condition $$ \lim_{t\to \infty }\frac{g(t)}{t}=1. $$ Examples illustrating the results are also given.http://ejde.math.txstate.edu/Volumes/2014/206/abstr.htmlFunctional differential equationstrongly decreasing solutionregularly varying functionslowly varying solution |
| spellingShingle | George E. Chatzarakis Kusano Takasi Ioannis P. Stavroulakis Precise asymptotic behavior of strongly decreasing solutions of first-order nonlinear functional differential equations Electronic Journal of Differential Equations Functional differential equation strongly decreasing solution regularly varying function slowly varying solution |
| title | Precise asymptotic behavior of strongly decreasing solutions of first-order nonlinear functional differential equations |
| title_full | Precise asymptotic behavior of strongly decreasing solutions of first-order nonlinear functional differential equations |
| title_fullStr | Precise asymptotic behavior of strongly decreasing solutions of first-order nonlinear functional differential equations |
| title_full_unstemmed | Precise asymptotic behavior of strongly decreasing solutions of first-order nonlinear functional differential equations |
| title_short | Precise asymptotic behavior of strongly decreasing solutions of first-order nonlinear functional differential equations |
| title_sort | precise asymptotic behavior of strongly decreasing solutions of first order nonlinear functional differential equations |
| topic | Functional differential equation strongly decreasing solution regularly varying function slowly varying solution |
| url | http://ejde.math.txstate.edu/Volumes/2014/206/abstr.html |
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